I'm looking to get games from other websites, this one ( http://www.ytv.com/games/98/freaky-friday/ ) in particular, onto my website. For some games, they have "Post this game on your website" buttons, and it allows you to post a bit of code on your site to embed it there. For ones that don't, like the one posted above, is there a way to embed it? Thanks!

You'd likely be infringing on a copyright, depending on the licensing of the game. I don't think that anyone here will give you all the steps you need to commit copyright infringement.

The game itself is a Flash applet. You'd have to learn how to copy and deploy Flash applets, as well as any related source code from the site that it may require, such as JavaScript. There's no guarantee that it would work.

might be a little late to reply, but I suggest not even looking into domain registration or web hosting....since ur new to web dev I assume. Just start learning and creating on ur localmachine for free.

Once you feel like ur ready then think about investing in shared hosting....something like this:
I've setup a basic website using justhost.com OR bluehost.com - i think they are the same comapny....With the cheapest tier it's like 5 bucks a month...and I think they give u 1 year domain registration with it if u sign up